How to duplicate an object with random start frame?
duplicate your character (shift+D) then right click to keep it in the same place above your original character

then shift the new duplicated character by changing the value of delta location/rotation/scale (object tab > delta transform)



now you have two distinct objects moving together, you can shift the keyframes of the second object in the dope sheet editor (b to box select all keyframes then g to move them) 



now you have two objects with two shifted animations.